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41
Designing Programs That Check Their Work
, 1989
"... A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It d ..."
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Cited by 307 (17 self)
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A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It designs program checkers for a few specific and carefully chosen problems in the class FP of functions computable in polynomial time. Problems in FP for which checkers are presented in this paper include Sorting, Matrix Rank and GCD. It also applies methods of modern cryptography, especially the idea of a probabilistic interactive proof, to the design of program checkers for group theoretic computations. Two strucural theorems are proven here. One is a characterization of problems that can be checked. The other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable, all problems in the class are checkable.
Tensor products of modules for a vertex operator algebras and vertex tensor categories
 in: Lie Theory and Geometry, in honor of Bertram Kostant
, 1994
"... In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announ ..."
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Cited by 44 (5 self)
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In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announcement has also appeared [HL1].
Homogeneous Designs and Geometric Lattices
 JOURNAL OF COMBINATORIAL THEORY SERIES A
, 1985
"... During the last 20 years, there has been a great deal of research concerning designs with A = 1 admitting 2transitive groups. The following theorems will be proved in this note; they are fairly simple consequences of the classification ’ of all finite simple groups (see, e.g., [6]). THEOREM 1. Let ..."
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Cited by 30 (4 self)
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During the last 20 years, there has been a great deal of research concerning designs with A = 1 admitting 2transitive groups. The following theorems will be proved in this note; they are fairly simple consequences of the classification ’ of all finite simple groups (see, e.g., [6]). THEOREM 1. Let 9 be a design with,! = 1 admitting an automorphism group 2transitive on points. Then 9 is one of the following designs: (i) PG(d qh (ii) AG(d, q), (iii) The design with u=q3 + 1 and k=q+ 1 associated with PSU(3, q) or *G,(q), (iv) One of two affine planes, having 34 or 36 points [S, p. 2361, or (v) One of two designs having u = 36 and k = 3 ’ [12]. THEOREM 2. Let Y be a finite geometric lattice of rank at least 3 such that Aut 6p is transitive on ordered bases. Then either (i) Y is a truncation of a Boolean lattice or a projective or affine geometry, (ii) 9 is the lattice associated with a Steiner system S(3, 6, 22), S(4, 7, 23), or S(5, 8, 24), or (iii) 6p is the lattice associated with the 65point design for PSU(3, 4). The groups in Theorems 1 and 2 are described in the course of the proof. It would, of course, be desirable to have more elementary proofs of both
Hyperelliptic jacobians without complex multiplication
 Math. Res. Letters
"... The aim of this note is to prove that in positive characteristic p ̸ = 2 the jacobian J(C) = J(Cf) of a hyperelliptic curve C = Cf: y 2 = f(x) has only trivial endomorphisms over an algebraic closure of the ground field ..."
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Cited by 25 (3 self)
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The aim of this note is to prove that in positive characteristic p ̸ = 2 the jacobian J(C) = J(Cf) of a hyperelliptic curve C = Cf: y 2 = f(x) has only trivial endomorphisms over an algebraic closure of the ground field
Sylow's Theorem in Polynomial Time
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1985
"... Given a set r of permutations of an nset, let G be the group of permutations generated by f. If p is a prime, a Sylow psubgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow psubgroup exists, and that for any two Sylo ..."
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Cited by 24 (8 self)
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Given a set r of permutations of an nset, let G be the group of permutations generated by f. If p is a prime, a Sylow psubgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow psubgroup exists, and that for any two Sylow psubgroups PI, P, of G there is an element go G such that Pz = g‘PI g. We present polynomialtime algorithms that find (generators for) a Sylow psubgroup of G, and that find ge G such that P, = g‘P, g whenever (generators for) two Sylow psubgroups PI, Pz are given. These algorithms involve the classification of all tinite simple groups. 0 1985 Academic Press. Inc. PART I 1.
Defect Zero pBlocks for Finite SIMPLE GROUPS
 Trans. Amer. Math. Soc
, 1996
"... . We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a pblock with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p\Gammablocks remained unclassified were the alternating groups An . Here we show that these ..."
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Cited by 19 (3 self)
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. We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a pblock with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p\Gammablocks remained unclassified were the alternating groups An . Here we show that these all have a pblock with defect 0 for every prime p 5. This follows from proving the same result for every symmetric group Sn , which in turn follows as a consequence of the tcore partition conjecture, that every nonnegative integer possesses at least one tcore partition, for any t 4. For t 17, we reduce this problem to Lagrange's Theorem that every nonnegative integer can be written as the sum of four squares. The only case with t ! 17, that was not covered in previous work, was the case t = 13. This we prove with a very different argument, by interpreting the generating function for tcore partitions in terms of modular forms, and then controlling the size of the coefficients usin...
The Status of the Classification of the Finite Simple Groups
 Mathematical Monthly
, 2004
"... Common wisdom has it that the theorem classifying the finite simple groups was proved around 1980. However, the proof of the Classification is not an ordinary proof because of its length and complexity, and even in the eighties it was a bit controversial. Soon after the theorem was established, Gore ..."
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Cited by 16 (0 self)
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Common wisdom has it that the theorem classifying the finite simple groups was proved around 1980. However, the proof of the Classification is not an ordinary proof because of its length and complexity, and even in the eighties it was a bit controversial. Soon after the theorem was established, Gorenstein, Lyons, and Solomon (GLS) launched a program to simplify large parts of the proof and, perhaps of more importance, to write it down clearly and carefully in one place, appealing only to a few elementary texts on finite and algebraic groups and supplying proofs of any “wellknown” results used in the original proof, since such proofs were scattered throughout the literature or, worse, did not even appear in the literature. However, the GLS program is not yet complete, and over the last twenty years gaps have been discovered in the original proof of the Classification. Most of these gaps were quickly eliminated, but one presented serious difficulties. The serious gap has recently been closed, so it is perhaps a good time to review the status of the Classification. I will begin slowly with an introduction to the problem and with some motivation. Recall that a group G is simple if 1 and G are the only normal subgroups of G; equivalently G ∼ =G/1 and 1 ∼ =G/G are the only factor groups
Primitive permutation groups of odd degree, and an application to finite projective planes
 MR878466 (88b:20007) Zbl 0606.20003 Nick Gill
, 1987
"... One of the most beautiful and important results concerning finite projective planes is the OstromWagner Theorem [26]: such a plane admitting a 2transitive collineation group must be desarguesian. It has long been conjectured that the same conclusion must hold if it is only assumed that there ..."
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Cited by 9 (2 self)
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One of the most beautiful and important results concerning finite projective planes is the OstromWagner Theorem [26]: such a plane admitting a 2transitive collineation group must be desarguesian. It has long been conjectured that the same conclusion must hold if it is only assumed that there
HR  A System for Machine Discovery in Finite Algebras
 ECAI 98 Workshop Programme
, 1998
"... We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theo ..."
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Cited by 8 (0 self)
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We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theory, and a look at HR in domains other than group theory.
Variables separated polynomials, the genus 0 problem and moduli spaces

, 2001
"... The monodromy method—featuring braid group action—first appeared as a moduli space approach for finding solutions of arithmetic problems that produce reducible variables separated curves. Examples in this paper illustrate its most interesting aspect: investigating the moduli space of exceptions to ..."
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Cited by 7 (2 self)
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The monodromy method—featuring braid group action—first appeared as a moduli space approach for finding solutions of arithmetic problems that produce reducible variables separated curves. Examples in this paper illustrate its most interesting aspect: investigating the moduli space of exceptions to a specific diophantine outcome. Explicit versions of Hilbert’s irreducibility theorem and Davenport’s problem fostered this technique and motivated the genus 0 problem started by J. Thompson and taken up by many group theorists. We review progress on the genus 0 problem in 0 characteristic, and its quite different contributions in positive characteristic. Example: Let f and h be polynomials with coefficients in a number field K. The classification of finite simple groups shows there is a bound on exceptional degrees for f to the following result. If f is indecomposable and h is not a composition with f, then f(x) − h(y) is irreducible. This answered challenge problems on factorization of variables separated polynomials posed by A. Schinzel in the early 60’s. This limitation result holds, however, only in